For (REF ), we apply solutions to the Bernstein approximation problem (see [1]} for a comprehensive survey). The original Bernstein approximation problem extends Weierstrass approximation to polynomial approximation in \(L^\infty (\mathbb {R},\mu )\)  for a measure \(\mu \)  that is absolutely continuous with respect to the Lebesgue measure. The work by Ditzian and Totik [2]}—which introduces moduli of smoothness—shows that tools used to solve the Bernstein approximation problem can also be useful for polynomial approximation in \(L^p(\mathbb {R},\mu )\)  for all \(p\ge 1.\)  We apply their results for the case \(p=2.\)
